# Basic Geometry : How to find the area of a 45/45/90 right isosceles triangle

## Example Questions

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Explanation:

### Example Question #1 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Consider an isosceles triangle with a height of 24 and a base of 12. What is the area of this triangle?

144

169

155

60

121

144

Explanation:

The formula for the area of a trianlge is A = base * height * (1/2).

We're lucky here, because the question gives us all of the values we need.  We simply need to plug them in:

A = base * height * (1/2) = 12 * 24 * (1/2) = 12 * 12 = 144

### Example Question #3 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Calculate the area of an isosceles right triangle who's hypotenuse is  inches.

Explanation:

The formula for the area of a triangle, right or not, is one half the base times height.

In this case, they are both  Therefore, the respective values are entered, yielding:

### Example Question #4 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

The side length of the 45-45-90 right triangle is , find the area of the right triangle.

Explanation:

The area of a triangle is:

where b=base, h=height, and A=area

### Example Question #5 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Find the area of the triangle below.

Explanation:

The key to finding the area of our triangle is to reaize that it is isosceles and therefore is a 45-45-90 triangle; therefore, we know the legs of our triangle are congruent and that each can be found by dividing the length of the hypotenuse by .

Rationalizing the denominator simplifies our result; however, we are interested in the area, not just the length of a leg; we remember that the formula for the area of a triangle is

where  is the base and  is the height; however, in our right triangle, the base and height are simply the two legs; therefore, we can calculate the area by substituting.

### Example Question #6 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Find the area of a  triangle that has a hypotenuse of .

Explanation:

To find the area of a triangle, we must use  where b=base and h=height.

In the problem, the only information given is what type the triangle is and what its hypotenuse is.

Given the area equation, the problem hasn't given any numbers that can be substituted into the equation to solve for an area. This means that the hypotenuse value must be used to determine the height and the base.

Because this is a 45/45/90 triangle, this means that it is also isosceles. Therefore, we can logic out that the base and the height must be the same.

The missing sides can be calulated in one of two ways:

1. Using the Pythagorean Theorem

2. Or using

If we were to use the Pythagorean Theorem, since we've already determined that b=h, that mean a=b in the equation. Let's say that

That means the Pythagorean Theorem can be rewritten as:

Now to substitute in the value of c to solve for the height and base.

Now that we have the base and the height, we can substitute the values into the area equation and get the triangle's area.

### Example Question #7 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Find the area of a  triangle with a hypotenuse of .

Explanation:

To find the area of a triangle, we must use  where b=base and h=height.

In the problem, the only information given is what type the triangle is and what its hypotenuse is.

Given the area equation, the problem hasn't given any numbers that can be substituted into the equation to solve for an area. This means that the hypotenuse value must be used to determine the height and the base.

Because this is a 45/45/90 triangle, this means that it is also isosceles. Therefore, we can logic out that the base and the height must be the same.

The missing sides can be calulated in one of two ways:

1. Using the Pythagorean Theorem

2. Or using

If we were to use the Pythagorean Theorem, since we've already determined that b=h, that means a=b in the equation. Let's say that

That means the Pythagorean Theorem can be rewritten as:

Now to substitute in the value of c to solve for the height and base.

Now that we have the base and the height, we can substitute the values into the area equation and get the triangle's area.

### Example Question #8 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of an isosceles right triangle is , what is the area of the triangle?

Explanation:

An isosceles right triangle is another way of saying that the triangle is a  triangle.

Now, recall the Pythagorean Theorem:

Because we are working with a  triangle, the base and the height have the same length. We can rewrite the above equation as the following:

Now, plug in the value of the hypotenuse to find the height for the given triangle.

Now, recall how to find the area of a triangle:

Since the base and the height are the same length, we can then find the area of the given triangle.

### Example Question #9 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of an isosceles right triangle is , what is the area of the triangle?

Explanation:

An isosceles right triangle is another way of saying that the triangle is a  triangle.

Now, recall the Pythagorean Theorem:

Because we are working with a  triangle, the base and the height have the same length. We can rewrite the above equation as the following:

Now, plug in the value of the hypotenuse to find the height for the given triangle.

Now, recall how to find the area of a triangle:

Since the base and the height are the same length, we can then find the area of the given triangle.

### Example Question #10 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of an isosceles right triangle is , what is the area of the triangle?

Explanation:

An isosceles right triangle is another way of saying that the triangle is a  triangle.

Now, recall the Pythagorean Theorem:

Because we are working with a  triangle, the base and the height have the same length. We can rewrite the above equation as the following:

Now, plug in the value of the hypotenuse to find the height for the given triangle.

Now, recall how to find the area of a triangle:

Since the base and the height are the same length, we can then find the area of the given triangle.

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